Scalar perturbations in...

Scalarperturbations inf(R)-cosmology

Jan Novak, MaximEingorn

Scalar perturbations in f(R)-cosmology

Jan Novak, Maxim Eingorn

Mathematical Institute of Academy of Sciences, Prague, Czech republicNorth Caroline Central University, Durham NC, U.S.A.

20.August 2013


Jan Novak, MaximEingornNEW TYPE OF ENERGY

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MODIFICATION OF THE THEORY OF GRAVITATION



S =1

2

g f (R) d4x + SM

SM = g LM d4x

T = 2LMg

+ gLM



We want to consider briefly the homogeneous (flat forsimplicity) background problem and establish allnecessary designations:Being based on the review [Antonio de Felice], let us startwith its equations (2.4) and (2.7):

F (R)R 12

f (R)g F (R) + gF (R) = 2T ,(2.4)

3F (R) + F (R)R 2f (R) = 2T , (2.7)

where F (R) = f (R), T = gT andF (R) = (1/

g)(

ggF ).



In the case of the spatially flat background spacetime withthe metrics

ds2 = gdxdx = dt2 + a2(t)(

dx2 + dy2 + dz2),

the Hubble parameter H = a/a and the scalar curvature

R = 6(2H2 + H).



these equations give the equations (2.15) and (2.16):

3FH2 = (FR f )/2 3HF + 2 , (2.15)

2FH = F HF + 2(+ P) , (2.16)

where the perfect fluid with the energy-momentum tensorcomponents T = diag(,P,P,P) satisfies thecontinuity equation

+ 3H(+ P) = 0 .



CONFORMAL NEWTONIAN GAUGE Now let us turn tothe formula (6.1) from [Antonio De Felice], describing theperturbed metric, and without loss of generality present itin the following form:

ds2 = (1 + 2)dt2 + a2(1 + 2)ijdx idx j .



Not taking into account the possible presence of thescalar field in equations (6.11)-(6.15) from [Antonio DeFelice], substituting = 0 and A = 3

(H

), we get a

particular system of equations. After further substitution = and = we get:



a2

+ 3H(

H + )

= 12F

[

(3H2 + 3H +

a2

)F

3H F + 3HF + 3F(

H + )

+ 2] ,

H + =1

2F

(F HF F

),F () = F ,

3(

H + H + )

+ 6H(

H + )

+ 3H +

a2=

=1

2F[3F+3H F6H2FF

a23F 3F

(H +

)

(

3HF + 6F)

+ 2(+ P)] ,



F + 3H F Fa2 1

3RF =

132( 3P)+

+F (3H + 3 + ) + 2F + 3HF 13

FR ,

F = F R,

R = 2[3(

H + H + )

+ 12H(

H + )

+

+

a2+ 3H 2

a2] .



So, the previous system of equations describes thescalar cosmological perturbations in the case of thenonlinear f (R) theory of gravity.


Jan Novak, MaximEingornLet us assume, that they permit the superposition

principle, then, neglecting P, we will solve them twice:1) dropping all terms containing the Laplace operator,

and substituting instead of and,2) substituting instead of and considering only one

point-like mass m, resting in the origin ofcoordinates)

Let us start by item 2), considering the domain r > 0(where = 0) and neglecting also R and, hence, F(that is assuming, that the scalarons mass is largeenough).



a2

+ 3H(

H + )

= 12F

[3HF + 3F

(H +

)],

H + =1

2F

(F

),

= 0 ,

3(

H + H + )

+ 6H(

H + )

+ 3H +

a2=

=1

2F[3F 3F (H + )

(3HF + 6F

)] ,

0 = F (3H + 3 + ) + 2F + 3HF ,

0 = 3(

H + H + )

+12H(

H + )

+

a2+3H2

a2.



= =

a

F

1a3

F+

3F2

4aF 2

F = 0 .



F (R) = 1 + o(1),

f (R) = R 2 + o(R R)



Thus, in the case of large enough scalarons mass wereproduce the linear cosmology from the nonlinearone, as it should be.



Taking the scalaron field into account, we need to solvethe following equation with respect to the scalar curvatureperturbation:

R(r) =3F

a2FR(r) +

1F2

a3c ,



So we have the solutions

=F

2F

Asinh(

a2F3F

r)

a2F3F

r+ B

cosh(

a2F3F

r)

a2F3F

r+

2

Fa3c

++

a,

= F

2F

Asinh(

a2F3F

r)

a2F3F

r+ B

cosh(

a2F3F

r)

a2F3F

r+

2

Fa3c

++

a,

where = GNm0

r

GNm02r30

r2 +3GNm0

2r0.



Neglecting the influence of the cosmologicalbackground, but not neglecting the scalaronscontribution, we have found the scalar perturbations.They represent the mix of the standard potential andthe additional Yukawa term.

=F

2F

2m12aF exp(

a2F3F

r)

r

2

Fa3c

+ a ,

= F

2F

2m12aF exp(

a2F3F

r)

r

2

Fa3c

+ a .

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